线段树基础操作

线段树 Segent Tree

基础操作

1.建树

2.增减某区间数值

3.增减混合乘除某区间数值

4.lazytag使用

5.区间求和

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 结构体

struct SegentTreeNode
{
    long long int value,lazytag;
};
SegentTreeNode SegentTree[400010];

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建树（一种lazytag）

void Build_tree(int root, int left, int right)
{
    SegentTree[root].lazytag = 0;//默认是加减，所以取0，表示当前根节点下的节点都要+0（虽然加和没加值的大小都一样）
    if (left == right) { SegentTree[root].value = Data[left]; return;}//已经到了最底层了，可以赋值了
    int mid = (left + right) >> 1;
    Build_tree(root << 1, left, mid );
    Build_tree(root << 1 | 1, mid + 1, right);
    SegentTree[root].value = SegentTree[root << 1].value + SegentTree[root << 1 | 1].value;
    return;
}

本质就是先往下一直找到最底层，然后递归赋值，一层一层往上面走

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增减某区间的值

void Search_Tree(int root,int left, int right,int visleft,int visright,long long  int add)//left和right是指当前root的左右范围，visleft和visright指要改动的区间
{
    if (right<visleft || left>visright) return;
    if (left >= visleft && visright >= right)//root的当前范围完全在要改区间内，也就是直接整体加减就足够了；
    {
        SegentTree[root].value += (long long int)(right-left+1)*add;//总共有几个，每个都要加add，所以一共加了这么多add；
        SegentTree[root].lazytag += add;//切记是+=，因为该root完全被覆盖，所以下面的暂时不用加，但是要记一下，之后如果调用的时候还是要加的
        return;
    }
    int mid = left + right >> 1;
    Push_down(root,left,right);//传下lazytag，避免下面有的值漏加了，导致二叉树出错
    Search_Tree(root << 1, left, mid, visleft, visright, add);
    Search_Tree(root << 1|1, mid+1, right, visleft, visright, add);
    SegentTree[root].value = SegentTree[root << 1].value + SegentTree[root << 1 | 1].value;//更新树的值
    return;
}

思路也还是一直先运算到最底层满足的，在从最底层往上递归更新值，就能实现改变底端值，上面的值也跟着改变的想法

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乘除某区间数值（只有乘除）

void Mulity_Tree(int root, int left, int right, int visleft, int visright, long long  int add2)//add2是要乘的值，mod是如果题目有要求取模才用，前面的值同上；
{
    if (right<visleft || left>visright) return;
    if (left >= visleft && visright >= right)
    {
        SegentTree[root].value = SegentTree[root].value*add2 %mod;//下面的值*n，那么上面的值也是*n，而且只要乘一次，和加法有区别
        SegentTree[root].lazytag2 = SegentTree[root].lazytag2 *add2%mod;//乘法须注意的点，往下乘的时候，加法的lazytag和乘法的lazytag值都要改变return;
    }
    int mid = (left + right) >> 1;
    Push_down(root, left, right);
    Mulity_Tree(root << 1, left, mid, visleft, visright, add2);
    Mulity_Tree(root << 1 | 1, mid + 1, right, visleft, visright, add2);
    SegentTree[root].value = (SegentTree[root << 1].value + SegentTree[root << 1 | 1].value) % mod;
    return;
}

思路和加减类似，要额外注意两个lazytag都要改变；

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加减混合乘除修改区间值

void Search_Tree(int root, int left, int right, int visleft, int visright, long long  int add)//加减部分
{
    if (right<visleft || left>visright) return;
    if (left >= visleft && visright >= right)
    {
        SegentTree[root].value = (SegentTree[root].value+(long long int)(right - left + 1) * add)%mod;
        SegentTree[root].lazytag = (SegentTree[root].lazytag+ add)%mod;
        return;
    }
    int mid = (left + right )>> 1;
    Push_down(root, left, right);
    Search_Tree(root << 1, left, mid, visleft, visright, add);
    Search_Tree(root << 1 | 1, mid + 1, right, visleft, visright, add);
    SegentTree[root].value = (SegentTree[root << 1].value + SegentTree[root << 1 | 1].value)%mod;
    return;
}
void Mulity_Tree(int root, int left, int right, int visleft, int visright, long long  int add2)//乘除部分
{
    if (right<visleft || left>visright) return;
    if (left >= visleft && visright >= right)
    {
        SegentTree[root].value = SegentTree[root].value*add2 %mod;
        SegentTree[root].lazytag2 = SegentTree[root].lazytag2 *add2%mod;//注意，在加减乘除混合的时候，乘除时要修改加减的lazytag和乘除的lazytag
        SegentTree[root].lazytag = SegentTree[root].lazytag * add2 % mod;
        return;
    }
    int mid = (left + right) >> 1;
    Push_down(root, left, right);
    Mulity_Tree(root << 1, left, mid, visleft, visright, add2);
    Mulity_Tree(root << 1 | 1, mid + 1, right, visleft, visright, add2);
    SegentTree[root].value = (SegentTree[root << 1].value + SegentTree[root << 1 | 1].value) % mod;
    return;
}

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lazytag使用

1）只有加减

void Push_down(int root,int left,int right)
{
    if (!SegentTree[root].lazytag) return;//如果加减的lazytag是0的话，就说明这个根节点下的值都是真实值，不需要额外更新
    long long int add = SegentTree[root].lazytag;
    int mid = (left + right) >> 1;
    SegentTree[root << 1].value += (long long int)(mid - left + 1) * add;
    SegentTree[root << 1|1].value += (long long int)(right - mid) * add;//改变左儿子和右儿子的值
    SegentTree[root << 1 ].lazytag += add;
    SegentTree[root << 1|1].lazytag += add;//把lazytag传下去
    SegentTree[root].lazytag = 0;
    return;
}

2）加减混合乘除（先乘除后加减原则）

void Push_down(int root, int left, int right)
{
    long long int add2 = SegentTree[root].lazytag2;
    long long int add = SegentTree[root].lazytag;
    int mid = (left + right) >> 1;
    SegentTree[root << 1].value = (SegentTree[root << 1].value*add2%mod+(long long int)(mid - left + 1) * add%mod)%mod;
    SegentTree[root << 1 | 1].value = (SegentTree[root << 1|1].value * add2%mod + (long long int)(right-mid) * add%mod ) % mod;
    SegentTree[root << 1].lazytag2 = SegentTree[root << 1].lazytag2*add2%mod;//先修改乘法的lazytag
    SegentTree[root << 1 | 1].lazytag2 = SegentTree[root << 1 | 1].lazytag2 * add2%mod;
    SegentTree[root << 1].lazytag = (SegentTree[root << 1].lazytag*add2+ add)%mod;//再修改加法的lazytag，修改加法的时候要记得有两步，既要乘也要加
    SegentTree[root << 1 | 1].lazytag = (SegentTree[root << 1|1].lazytag * add2 + add) % mod;
    SegentTree[root].lazytag = 0;
    SegentTree[root].lazytag2 = 1;
    return;
}

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区间求和

long long int  Find_sum(int root, int left, int right, int visleft, int visright)
{
    if (right<visleft || left>visright) return 0;//不属于范围内的就+0
    if (left >= visleft && right <= visright) return SegentTree[root].value;
    Push_down(root,left,right);//在读取或者修改的时候都要传lazytag，为了读取到真实的实时值
    int mid = (left + right) >> 1;
    return Find_sum(root << 1, left, mid, visleft, visright)+ Find_sum(root << 1 | 1, mid + 1, right, visleft, visright);//
   
}